Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 b Solution Created 2026-10-03 Updated 2026-10-06
- Vertical chemical quenching: when the eddy mixing time becomes shorter than the chemical relaxation time, transported gas retains a deeper abundance rather than reaching local thermochemical equilibrium. For example, carbon monoxide–methane quenching can maintain excess carbon monoxide in cool upper giant-planet gas where equilibrium favors methane.
- Atmospheric photochemistry: ultraviolet photons dissociate molecules and drive reaction networks away from thermal equilibrium. For example, irradiation of CH4-containing gas can produce acetylene and hydrocarbon precursors of an atmospheric haze.
- Horizontal chemical quenching: winds cross a day-night temperature gradient faster than reactions can reset the composition. For example, CO-rich dayside gas can reach a cooler hot Jupiter nightside without converting most of its carbon to CH4.
These are three distinct ways to maintain disequilibrium chemistry in an exoplanet atmosphere; each compares chemistry with a different transport or irradiation process.
Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 2 a Solution Created 2026-10-03 Updated 2026-10-14
For a representative hydrogen-rich hot Jupiter, increasing altitude lowers pressure, slows collision-driven chemistry, and increases exposure to stellar UV. Three principal regimes follow.
At high pressures, illustratively -- bar, collisions and sufficiently high temperature make thermochemical equilibrium a useful approximation. Molecular abundances minimize the free energy subject to elemental conservation. Condensation and atmospheric condensate rainout can remove selected elements from the gas.
At intermediate pressures, illustratively bar, vertical mixing and horizontal winds can outrun chemical relaxation time. The chemical quench level is defined by ; above it, some abundances retain values from a deeper or hotter region. Horizontal chemical quenching similarly occurs when chemical adjustment is slower than advection between the day and night hemispheres.
At low pressures, illustratively bar, atmospheric photochemistry becomes important: photodissociation and radical reactions change equilibrium abundances and can produce hydrocarbons and atmospheric hazes. At still lower pressures, roughly bar and above in altitude, photoionization and an escaping thermosphere can become important.
Nominal deep-equilibrium, transport-quenching and photochemical regimes in a hydrogen-rich irradiated atmosphere
. Details. These pressures label a representative sketch, not universal interfaces. UV optical depth, stellar spectrum, gravity, metallicity, reaction rates and eddy diffusivity determine the transitions; quenching is species-dependent.
Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 4 j Solution Created 2026-10-03 Updated 2026-10-06
Three solar-system effects illustrate chemical disequilibrium. First, vertical mixing carries carbon monoxide from hot deep regions into cooler observable layers of giant planets, where its local-equilibrium abundance would be small. Second, phosphine survives in the observable atmospheres of Jupiter and Saturn because deep supply competes with its destruction, making it a tracer of transport. Third, methane photolysis creates hydrocarbons such as ethane and acetylene, and can help form upper-atmospheric atmospheric hazes in giant planets and Titan.
Three exoplanet effects are complementary. First, carbon monoxide–methane quenching can keep carbon monoxide abundant and methane depleted in cool observable gas, including hot-Neptune/giant atmospheres that would locally favor methane. Second, atmospheric photochemistry dissociates methane/ammonia and produces radicals, hydrogen cyanide, hydrocarbons and possibly atmospheric hazes, changing spectral bands from their equilibrium strengths. Third, horizontal chemical quenching can preserve hot-dayside carbon chemistry on the cooler nightside/terminator and reduce the abundance contrast expected from separate local-equilibrium columns.
Transport, photon-driven reactions and finite chemical adjustment times change the observable composition. The effects are conditional on the species' chemical timescale relative to mixing, advection and irradiation; they do not imply that every molecule is out of equilibrium everywhere.
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 2 c iii Solution Created 2026-10-03 Updated 2026-10-06
Convection mixes material vertically where the Schwarzschild criterion or Ledoux criterion allows it. Its chemical effect is controlled by competition between transport and reaction, rather than by a direct alteration of the equilibrium constant. Parametrize vertical transport by a vertical eddy diffusion coefficient ; the eddy mixing time over a scale height isLet be the chemical relaxation time of the relevant interconversion. If reactions are faster, the gas follows local thermochemical equilibrium. If transport becomes faster, composition can be carried upward faster than it re-equilibrates. A chemical quench level is approximately where the two times match:For example, hot deep gas may contain abundant carbon monoxide. Strong mixing can maintain it in cooler layers where equilibrium would favour methane; this is carbon monoxide–methane quenching. Similar reasoning applies to nitrogen–ammonia quenching. The cooler nightside commonly has slower reaction rates and can preserve such quenched composition more readily, whereas sufficiently rapid dayside chemistry can restore local equilibrium. Neither statement is universal without the actual reaction network and temperature.
The observable atmosphere of a strongly irradiated hot Jupiter may be radiative and stable even if the deeper interior is convective. Deep convection influences the photosphere only if some mixing crosses the intervening radiative region, for example through overshoot or other dynamical transport. Moreover, transport from day to night is horizontal circulation, not vertical convection; its relevant competition is atmospheric advection time versus chemical relaxation time, giving horizontal chemical quenching. Convection can affect either hemisphere indirectly through connected vertical transport; strong deep convection alone does not guarantee a quenched observable composition.
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 2 c ii Solution Created 2026-10-03 Updated 2026-10-06
The two measurements sample different atmospheric regions and different radiative transfer geometries. An exoplanet emission spectrum probes the dayside; an exoplanet transmission spectrum probes the morning and evening limbs of the day-night terminator. Several alternatives to an exoplanet cloud deck can therefore suppress the transmission feature.
- A cooler terminator has a smaller atmospheric scale height , so the same change in slant optical depth produces a smaller transit-area change. A larger mean molecular weight has the same geometric effect. The assumption that the nightside temperature sets both limbs may itself be wrong.
- Genuine chemical heterogeneity can reduce terminator carbon monoxide. At lower temperature, thermochemical equilibrium can favour methane through , if local reaction times are sufficiently short. Atmospheric photochemistry, elemental abundance differences, and horizontal chemical quenching can modify this simple picture.
- The feature depends on a band-to-continuum opacity ratio, not abundance alone. A continuum enhanced by other gas absorption or a partly covered heterogeneous limb can mute the feature; the two-limb transmission spectrum illustrates this averaging. Stellar surface heterogeneity and measurement systematics can also bias an inferred feature amplitude.
- A strong CO emission feature can be amplified by an atmospheric thermal inversion, because the line forms in hotter gas than the continuum. It is consequently not an unambiguous abundance measurement without a simultaneous atmospheric pressure-temperature profile fit.
Temperature, mean molecular weight, limb chemistry, continuum opacity and the dayside temperature gradient are alternatives or degeneracies to test. In particular, the three-scale-height amplitude alone cannot establish different CO abundances between the hemispheres.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 3 a Solution Created 2026-10-03 Updated 2026-10-05
Four mechanisms that can produce disequilibrium chemistry in an exoplanet atmosphere are:
- Atmospheric photochemistry. Stellar ultraviolet photons initiate reactions whose products need not follow local thermochemical equilibrium. The ozone layer on Earth is a solar-system example. Calculations for HD 189733 b predict enhanced hydrogen cyanide and acetylene from the processing of methane and ammonia; these are model examples rather than assertions of an unambiguous detection.
- Vertical transport and chemical quenching. When the eddy mixing time is shorter than the chemical relaxation time, gas retains a deeper abundance above its chemical quench level. The excess carbon monoxide in Jupiter's cool atmosphere exemplifies carbon monoxide–methane quenching. Models of HD 189733 b predict quenched methane and ammonia abundances differing from their local chemical equilibrium values. The enhancement or depletion depends on the underlying atmospheric pressure-temperature profile.
- Horizontal chemical quenching. If the atmospheric advection time is short, winds move chemically processed gas into regions with different irradiation or temperature faster than it can re-equilibrate. Transport of gas within Earth's ozone layer moves material away from its local photochemical production regions. Models of HD 209458 b show that dayside carbon monoxide-rich composition can persist into the cooler nightside instead of forming the local chemical equilibrium amount of methane.
- Condensation with sedimentation or rainout. Finite-rate cloud formation can depart from phase equilibrium, while atmospheric condensate rainout removes elements from a layer and changes its gas composition. Earth's water atmospheric cold trap limits the supply of water to the stratosphere. In HD 209458 b models, titanium-bearing condensates can settle and suppress upper-atmospheric titanium monoxide. The remaining gas can still be in local chemical equilibrium with its depleted inventory: rainout is an open-column effect, not necessarily a failure of equilibrium among all gas reactions.
The exoplanet transport and photochemical examples follow kinetic atmosphere calculations and models including horizontal transport; the condensate example is examined in cold-trap calculations.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 3 b i Solution Created 2026-10-03 Updated 2026-10-14
Take the logarithm to base ten and write pressure in units of , so its argument is dimensionless. Continuity of the atmospheric pressure-temperature profile givesThusIf the logarithm means , the equivalent constant is . A plot of temperature against logarithmic pressure is vertical in each isothermal region and straight between the two endpoints.
Atmospheric pressure-temperature profile
. Details. The processes can be organized by the supplied pressure ranges, although the exact boundaries require reaction rates, irradiation and mixing information:
- At , the dense gas can approach thermochemical equilibrium because collisions and reactions are relatively rapid. Deep carbon monoxide and molecular nitrogen can provide reservoirs for transported material.
- At , the falling temperature slows chemical conversion. Vertical transport can produce a chemical quench level when the chemical relaxation time crosses the eddy mixing time. Horizontal chemical quenching is also possible if dayside and nightside conditions differ. Suitable species can condense and undergo atmospheric condensate rainout where a saturation curve is crossed.
- At , slow thermal chemistry permits a quenched atmospheric mixing ratio to survive. Atmospheric photochemistry can dominate where stellar ultraviolet photons penetrate, often at still lower pressures; atmospheric haze may form from its products. Extremely high layers can also experience atmospheric escape.
The profile identifies plausible chemical regimes, but does not fix their transition pressures by itself. In particular, cloud formation depends on the species-specific condensation curve, and ultraviolet processing depends on shielding.

